Extending Grundy domination to $k$-Grundy domination

The Grundy domination number of a graph $G=(V,E)$ is the length of the longest sequence of unique vertices $S=(v_1,\dots ,v_k)$ satisfying $N[v_i]\setminus {\cup }_{j=1}^{i-1}N[v_j]\mathrm{\ne }\mathrm{\varnothing }$ for each $i\in [k]$. Recently, a generalization of this concept called $k$-Grundy domination was introduced. In $k$-Grundy domination, a vertex $v$ can be included in $S$ if it has a neighbor $u$ such that $u$ appears in the closed neighborhood of fewer than $k$ vertices of $S$. In this paper, we determine the $k$-Grundy domination number for some families of graphs, find degree-based bounds for the $k$-$L$-Grundy domination number, and define a relationship between the $k$-$Z$-Grundy domination number and the $k$-forcing number of a graph.